Park Street was located between Cherry Street and Union Streets. Cherry St was later renamed Chestnut Street after the rerouting. It survives as an alleyway and parking lot until it reaches Squid Row, another perpendicular alley that parallels Center and Chestnut Streets. In the Google StreetView below, Park Street appears as the driveway/alley to the right of the lime green car. The tracks meet the street at roughly this spot after heading out from the tunnel, which is at left.
Based on the location of the flag light, the cement base of which still survives about five yards from where the tracks merge with the street, it is probable that the train exited the tunnel and then stopped to pick up people at Park Street. My hunch is that this stop was primarily used by people heading toward Los Gatos who wanted to bypass the mess of tourists at Santa Cruz Depot. The train must have stopped in the middle of Cherry and Chestnut Streets to pick up passengers since there was not much space coming out of the tunnel for passengers to board. There is a fairly long stretch of clearing on the east side exit of the tunnel near Park Street and what used to be Cherry Street. Today it is a driveway (below) but it may have once had a platform of some type.
The flag-stop does not merit a mention in the 1899 Station Book nor any notice on the various timetables. It was located roughly 80 mile south of San Francisco. Nothing else is currently known or written about this obscure flag stop.